A357472 Decimal expansion of the real root of x^3 + x^2 + 2*x - 1.

3, 9, 2, 6, 4, 6, 7, 8, 1, 7, 0, 2, 6, 4, 0, 8, 1, 1, 7, 6, 4, 8, 7, 9, 5, 9, 4, 8, 8, 4, 3, 4, 1, 2, 5, 0, 7, 0, 3, 7, 6, 4, 9, 6, 8, 5, 9, 3, 4, 8, 2, 5, 8, 9, 7, 3, 1, 1, 3, 9, 6, 4, 9, 8, 4, 4, 5, 1, 7, 1, 6, 6, 8, 4, 7, 0, 8

Offset: 0

Author

Wolfdieter Lang, Oct 25 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 - 1/3 where r0 is the real root of y^3 + (5/3)*y - 43/27.

The other roots of x^3 - x^2 + 2*x - 1 are (-1 + w1*((43 + 9*sqrt(29))/2)^(1/3) + ((43 - 9*sqrt(29))/2)^(1/3))/3 = -0.6963233908... + 1.4359498641...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 is one of the complex conjugate roots of x^3 - 1.

Using hyperbolic functions these roots are -(1 + sqrt(5)*(sinh((1/3)*arcsinh( (43/50)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((43/50)*sqrt(5)))*i))/3, and its complex conjugate.

Examples

			0.3926467817026408117648795948843412507037649685934825897311396498445171668...
		

Crossrefs

Cf. A160389, A255524, A255249, A357470, A357471.

Programs

Maple

Digits:=100: u := 2/(43 + 9*sqrt(29)): (-5*u^(1/3) + u^(-1/3) - 1)/3:
evalf(%*10^78): ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 01 2022

Mathematica

RealDigits[x /. FindRoot[x^3 + x^2 + 2*x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 26 2022 *)

Formula

r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) - 20*(4*(43 + 9*sqrt(29)))^(-1/3))/6.

r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) + w1*(4*(43 - 9*sqrt(29)))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i) is one of the complex conjugate roots of x^3 - 1.

r = (-1 + 2*sqrt(5)*sinh((1/3)*arcsinh((43/50)*sqrt(5))))/3.

r = (-5*u^(1/3) + u^(-1/3) - 1)/3 where u = 2/(43 + 9*sqrt(29)). - Peter Luschny, Nov 01 2022

r = (-1/3) + (43/45) * Hyper2F1([1/3, 2/3], [3/2], -43^2/(5*10^2)). - Gerry Martens, Nov 04 2022